This thesis addresses the generation and propagation of waves in Earth's fluid outer core, with the objective of better understanding their role in maintaining our planet's magnetic field. The classes of waves studied, which owe their existence both to the rapid rotation of the Earth about its axis and the vibration of magnetic field lines, are united by the fact they are dispersive - which is to say, the velocity at which a given wave travels is a function of the spacing and orientation of its wave crests; the behaviour of such waves is remarkably rich and has frequently counter-intuitive consequences.

Our analysis begins at the very smallest scales of the convection which stirs the liquid iron outer core, asking how localised features - such as turbulent eddies or buoyant parcels - might incite oscillatory motions at this fundamental scale. Of particular interest are the mechanisms by which columnar flow structures are established and sustained. We show that, in a situation where the wave source is threaded by a large-scale magnetic field, these processes are managed by a previously-overlooked denomination of oscillations dubbed inertial-Alfvén waves.

These objects establish the starting point for an investigation into the journeys embarked upon by packets of waves throughout the outer core, following their progress as global-scale variations in the magnetic field force them to evolve both their spatial structure and propagation velocity. A growing ambient field slows them considerably, even to the point where certain rays are arrested completely and quenched by electromagnetic losses. It is seen that the whole spectrum of small-scale waves influenced by rotation and mean magnetic field have a part to play in this story.

Finally, we study a rather distinct variety of waves: hydrodynamic quasi-geostrophic Rossby waves, which rely upon the rapid background rotation and the slope of the core-mantle boundary for their existence. Our interest in these waves is as a possible explanation for the westward drift of the non-dipolar part of the observable magnetic field. As it happens, the crests of these waves invariably progress eastward; however, for a certain subset - corresponding to radially-extended sheet-like flows compatible with the convective driving - they can nevertheless convey energy to the west. This insight raises an intriguing new possibility in the mission to solve this centuries-old puzzle.